What is density of dark matter as a function of distance from the galactic core?

[Lotto]

Homework Statement

Because of dark matter, our stars in the Galaxy with core of radius $r_1$, whose matter is distributed spherically symetrically, are moving with speed $v_0$ that is not dependent on distance from the Galaxy's centre. This is true for distances from the centre smaller than $r_2$. Consider the distribution of dark matter to be spherically symetric around the centre. What is denisty of dark matter dependent on distance $r$ from the Galaxy's centre in interval from $r_1$ to $r_2$?

Relevant Equations

$G\frac{mM}{r^2}=m\frac {{v_0}^2}{r}$
$M_1=\frac{{v_0}^2r_1}{G}$

This problem builds on my previous post, where we calculated that core's mass is $M_1=\frac{{v_0}^2r_1}{G}$. So if we consider mass of dark matter dependent on distance $r$ to be $M_2(r)$, we can calculated it from

$G\frac{(M_2(r)+M_1)m}{r^2}=m\frac{{v_0}^2}{r}.$

So $M_2(r)=\frac{{v_0}^2}{G}(r-r_1)$.

An average density in interval from $r$ to $r+\Delta r$ is $\frac{\frac{{v_0}^2}{G}(r+\Delta r-r_1)-\frac{{v_0}^2}{G}(r-r_1)}{\frac 43 \pi [(r+\Delta r)^3-{r_1}^3]-\frac 43 \pi (r^3-{r_1}^3)}$. If we make a limit of it for $\Delta r \rightarrow 0$, we get

$\rho(r)=\frac{{v_0}^2}{4\pi r^2G}$.

Is it correct? If not, where are my thoughts wrong?

 

[mfb]

Looks right.